There is a big literature on this. Typing 'circle fitting' into google with give you a lot of resources. Most of the approaches I know of deal with estimating both the center and the radius, but they could easily be adapted to estimate just the center if that is what you want. Your question is probably more appropriate for CrossValidated stats.stackexchange.com/questions. You might have better luck there.

Oh yes! I much prefer Herman Wouk's ryhming version ''When in danger or in doubt, run in circles, scream and shout'' anyway. Thanks!

I think, due to the symmetry, it essential does act in a linear way. I'm pretty sure I can describe very accurately how well your estimator will work. I'll post this as an(other) answer a bit later.

I will Will, I will. Just give me a little time.

Actually, I lie. In this case, I think this is exactly what you want to do! I recommend this answer be marked as correct. My answers can just be considered as advertising for the important and interesting the field of circular statistics.

That's a good answer Niels, and I believe it will work quite well in this case. One trouble I see is that by squaring you have multiplied the 'noise' (if you like) by 2. So, I think that this estimator will be consistent, but not efficient.

@Andrei: If you have further questions, feel free to send me an email.

Hi Will, I use metapost. The actually data for the pdf (it's a weighted sum of von Mises distributions by the way) comes from a java library (unfortunately) and then the plotting gets done by metapost. At some point I plan on releasing all of the code I have for simulations and plotting under the CRAPL matt.might.net/articles/crapl. However, at the moment I feel the code is even too crap for the CRAPL :(

@Pablo: Thanks very much for this! This is a definitely a neat and simple example.

Nice answer! Is it true that your vector $v$ is an example of what is sometimes called a `badly approximable vector'?

I actually just got back from a run when I read this question. I almost want to go for another run to experiment :)